To rewrite the expression $\sin^2(5x) \cos^2(5x)$ in terms of first powers of cosine, we'll use the power-reducing identities.

Step 1: Apply the power-reducing identity for $\sin^2$ and $\cos^2$

The power-reducing formulas are:

$\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$ $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$

Step 2: Express $\sin^2(5x)$ and $\cos^2(5x)$

We can use the identities for both $\sin^2(5x)$ and $\cos^2(5x)$:

$\sin^2(5x) = \frac{1 - \cos(10x)}{2}$ $\cos^2(5x) = \frac{1 + \cos(10x)}{2}$

Step 3: Multiply the two expressions

Now, multiply $\sin^2(5x)$ and $\cos^2(5x)$:

$\sin^2(5x) \cos^2(5x) = \left( \frac{1 - \cos(10x)}{2} \right) \left( \frac{1 + \cos(10x)}{2} \right)$

Step 4: Simplify using the difference of squares

Using the difference of squares formula, $(a - b)(a + b) = a^2 - b^2$, we get:

$\sin^2(5x) \cos^2(5x) = \frac{(1)^2 - (\cos(10x))^2}{4}$

Simplify the expression:

$\sin^2(5x) \cos^2(5x) = \frac{1 - \cos^2(10x)}{4}$

Step 5: Further simplification

We can use the Pythagorean identity $\sin^2(\theta) = 1 - \cos^2(\theta)$ to simplify this further:

$\sin^2(5x) \cos^2(5x) = \frac{\sin^2(10x)}{4}$

So, the expression $\sin^2(5x) \cos^2(5x)$ in terms of the first power of $\cos$ is:

$\sin^2(5x) \cos^2(5x) = \frac{\sin^2(10x)}{4}$

Let me know if you'd like more details or if you have any questions!

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Here are 5 questions that expand on this topic:

1. How can I rewrite expressions involving other trigonometric identities using power-reducing formulas?
2. What are other ways to simplify trigonometric expressions involving both sine and cosine?
3. How do the power-reducing formulas relate to the double-angle identities?
4. What happens if we need to reduce expressions that involve higher powers of sine or cosine, like $\sin^4(x)$ or $\cos^6(x)$?
5. How do these formulas change when the angle is multiplied by a factor larger than 2, like in $\sin^2(3x)\cos^2(3x)$?

Tip: Power-reducing formulas are often used to simplify integrals involving trigonometric functions, especially when you're trying to express everything in terms of a single trigonometric function.